matita/matita/contribs/lambdadelta/static_2/syntax/sh_lt.ma

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   3 (*      ||M||                                                             *)
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  14
  15 include "static_2/syntax/sh_props.ma".
  16
  17 (* SORT HIERARCHY ***********************************************************)
  18
  19 (* strict monotonicity condition *)
  20 record sh_lt (h): Prop ≝
  21 {
  22   next_lt: ∀s. s < ⫯[h]s
  23 }.
  24
  25 (* Basic properties *********************************************************)
  26
  27 lemma nexts_le (h): sh_lt h → ∀s,n. s ≤ (next h)^n s.
  28 #h #Hh #s #n elim n -n [ // ] normalize #n #IH
  29 lapply (next_lt … Hh ((next h)^n s)) #H
  30 lapply (le_to_lt_to_lt … IH H) -IH -H /2 width=2 by lt_to_le/
  31 qed.
  32
  33 lemma nexts_lt (h): sh_lt h → ∀s,n. s < (next h)^(↑n) s.
  34 #h #Hh #s #n normalize
  35 lapply (nexts_le … Hh s n) #H
  36 @(le_to_lt_to_lt … H) /2 width=1 by next_lt/
  37 qed.
  38
  39 axiom sh_lt_dec (h): sh_lt h → sh_decidable h.
  40
  41 axiom sh_lt_acyclic (h): sh_lt h → sh_acyclic h.